Algorithme pour la résolution des systèmes flous

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Algorithme pour la résolution des systèmes flous

Michel Prevot

To cite this version:

Michel Prevot. Algorithme pour la résolution des systèmes flous. [Research Report] Institut de

mathématiques économiques (IME). 1978, 11 p. �hal-01527191�

N°29

ALGORITHME POUR LA RESOLUTION DES SYSTEMES FLOUS

1 Michel PREVOT

Mai 1978

Le but de cette Collection est de diffuser rapidement une première version de travaux afin de provoquer des

.

discussions scientifiques. Les lecteurs désirant entrer en rapport avec un auteur sont priés d'écrire à l'adresse suivante: INSTITUT DE MATHEMATIQUES ECONOMIQUES

4 - Boulevard Gabriel - 21000 Dijon " France

Cette étude fera l'objet d'une communication au Colloque International sur la Théorie et les Applications des Sous- Ensembles Flous, Marseille, 20-22 septembre 1978.

TRAVAUX DEJA PUBLIES

N"1 Michel PREVOT: Théorème du point fixe. Une étude topologique générale(juin 1974) N°2 Daniel LEBLANC; L'introduction des consommations intermédiaires dans le

modèle de LEFEBER (juin 1974)

N°3 Colette BOUNON: Spatial Equilibrium of the Sector in Quasi-Perfect Compétition (september 1974)

N°4 Claude PONSARD: L'imprécision et son traitemebt en analyse économique (septembre 1974)

?5 Claude PONSARD: Economie urbaine et espaces métriques (septembre 1974) N°6 Michel PREVOT: Convexité (mars 1975)

N°7 Claude PONSARD: Contribution à une théorie des espaces économiques imprécis (avril 1975)

N°8 Aimé VOGT: Analyse factorielle en composantes principales d'un caractère de dimension - n (juin 1975)

N°9 Jacques THISSE et Jacky PERREUR: Relation between the Point of Maximum Profit and the Point of Minimum Total Transportation Cost:

A Restatement (juillet 1975)

N°10 Bernard FUSTIER: L'attraction des points de vente dans des espaces précis et imprécis (juillet 1975)

N°11 Régis DELOCHE: Théorie des sous-ensembles flous et classification en analyse économique spatiale (juillet 1975)

N°12 Gérard LASSIBILLE et Catherine PARRON: Analyse multicritère dans un contexte imprécis (juillet 1975)

N°13 Claude PONSARD: On the Axiomatisation of Fuzzy Subsets Theory (july 1975) N°14 Michel PREVOT: Probability Calculation and Fuzzy Subsets Theory (August 1975) N°15 Claude PONSARD: Hiérarchie des places centrales et graphes flous

(avril 1976)

N°16 Jean-Pierre AURAY et Gérard DURU: Introduction à la théorie des espaces multiflous (avril 1976)

fd°17 Roland LANTNER, Bernard PETITJEAN, Marie-Claude PICHERY: Jeu de simulation du circuit économique (Août 1976)

N°18 Claude PONSARD: Esquisse de simulation d'une économie régionale: l'apport de la théorie des systèmes flous (septembre 1976)

N°19 Marie-Claude PICHERY: Les systèmes complets de fonctions de demande (avril 1977 N°20 Gérard LASSIBILLE et Alain MINGAT: L'estiamtion de modèles à variable

dépendante dichotomique. La sélection universitaire et la réussite en première année d'économie (avril 1977)

N°21 Claude PONSARD: La région en analyse spatiale (mai 1977)

N°22 Dan RALESCU: Abstract Models for Systems Identification (juin 1977) N°23 Jean MARCHAL et François POULON: Multiplicateur, graphes et chaines de

' Markov (décembre 1977)

N°24 Pietro BALESTRA: Déterminant and Inverse of a Sum of Matrices with Applica- tions in Economics and Statistics (avril 1978)

N°25 Bernard FUSTIER: Etude empirique sur la notion de région homogène (avril 1978) N°26 Claude PONSARD: On the Imprecision of Consumer's Spatial Preferences

(avril 1978)

N°27 Roland LANTNER: L'apport de la théorie des graphes aux représentations de l'espace économique (avril 1978)

N°28 Emmanuel JOLLES: La théorie des sous-ensembles flous au service de la décision: deux exemples d'application.(rnai 1978)

Sanchez formulated conditions and theoretical methods to resolve fuzzy relations. The purpose of this study is to give an algorithm which would actual- ly enable us to determine the functions of appartenance of unknown relations.

� and � being known, we have to détermine �' such that

Let us notice that, given the operation symmetry of o min, and since %£, e are known,�can be determined in the same way.

It suffices to

formxl. (y, zl), the order columns of J*-

are determined in a similar way

- write the values of

J^S^ in a table and put the values of J*-^ (x, zl) into columns

- transform the table by replacing

/6L(x, y) by zero if )Xq (x, y) ��lze (x, zl) }.\Sl(x, y) by one if^g^ (x, y) ��À�e (x, zl) - choose the smallest value of �,» ce (x, zl) let

}'exo' zl) = 0{

If the corresponding line is s that o�PP.5�(x. , y) = 0 V

y1the problem is impossible and the calculation stops.

If the corresponding line contains 1 at least once,let

yj'" yl be the values of y such that �s(,(X, y) = 1.

We choose one of the values }l;j (x, y.) )^^(x' y,) equal to 1 and all the other values equal to orles than *( .

We cross out the correspondin9 column (or colums) and we use the reduced table to iterate the process taking the values of J.'�(x, zl)

increasing by.

In the various iterations three cases may occur

i ) we reach a line madeYf zeros, the problem is impossible.

ii ) all the columns are crossed out before reaching the highest value of � � , the probleme is impossible

i i i the line corresponding to the highest value of u g�ccntains non crosse�1 's ; the problem has at least one solution.

Justification of the algorithm

- 2 -

3 x0 such that Vy£ F

�� (x0, y)

` ?4� (XO, zl).

We have seen that the problem had no solution

If the line contains value one, at least once, we can classify the s into two subsets

�hatever the value given to

)-Id . The equally condition is never fulfilled.

We reach an impossibility

The condition is written

Io satisty this condition it is sutticient that

For all the other values

For the rest of the calculation we can cross out the corresponding columns. Indeed

- 3 -

Let us study the last three cases

out i) we reach a line only containing non crossedKzeros, as the crosse�alues play no part. and as for ail the other values

The problem is impossible

ii) All the columns are crossed out : we have seen that ^ y 6 F

The problem is impossible

iii) We can form all the values of

À 8 and the problem has at least one solution.

Examples :

1

impossible problem

This is impossible

From the first line we obtain

P 3 (y3' zl) = 0,4

- 5 -

equality.

The problem is impossible )

zl) = 0,6

equality

There is an infinite number of solutions to this problem

r _ t

In all cases thèse solutions belong to

- 6 -

one equality

equality.

The solutions are

C r-

In all cases the solutions are belong to the set

2.) Values

Me(x, zl) are all distinct There exist values

^(x, y) =

'-C(x, zl ) write down the values

of P g) and add a column giving the values of )-k ce (x, zl)

transform the table by replacing

- 7-

conserving the value if

(x, y) =

)J(;'(x, zl)

We use the algorithm of 1 until we reach a line with values of ^

^ different from 0 and 1.

Two cases are to be distinguished

i) In this line there are now crossedY15 ; replace

f'5{(XI

)=

y� (x, zl) by zero and go on with the algorithm.

ii) This line contains only non crossed)1êros cross out the line and mark the corresponding column.

Go on with the algorithm until the bottom of the table.

The solutions are obtained as expected but it thould be noted that,if a alune

of

(y, zl) belongs to a marked column its value must at least be equal to Y,

P-1 (x, y) =

iu°P (x, z1).

Justification of the algorithm The only case to study is line

We have seen that the crosse�vfi'ümbers play no part.

As

he

' s i ncrease

But we must take into account

We must choose

�3� (x,

Y2) plays no part and can be replaced by zero.

By assumption y£ (x, YI) =

2 (x, zl) o t

The crossed

To fulfill the condition it suffices that

- 8 -

If y y is determined later by calculation» thé )À3's increase J^iS (YI' z)

\ /

**£

(x, zl)

then the condition is automatically satisfied.

If

ft (y,, zl) is not defined, the restriction imposed on its varia- tion implies that:

and we can remove the line.

Examples : 1)

The solutions are

They belong to

- 9 -

The solutions are (0,6, 0,5,

0, l], 0,4) They are belong to (0,6, 0,5, 1 , 0,4)

The solutions

are 11 0 - 1] , fo- il, 0,6,

0,6 j They are belong to

0,6 j

- 10 -

The solutions are

0,6 � 0 - 13 , [ 0, 5, 1] , 0,4 3 They belong to

�

0,6, 1, 1,

0,4]

7

3) The values

are not all distinct

We begin thé algorithm as in (1).

We come to two equal values of /

g(x, zl) =

J

*

(x', zi) Two cases are to be distinguished

. - -- r

_ .... - 1

If Y1 n

Y2 = 0 we go on with thecalculation without taking the equality of the values of

into account.

We take ^^ (y,, zl) = fqj (x, zl) and we take higher values.

Justification of the algorithm :

If YI n Y2 = 0 to détermine - J

everything happer as if the }!

's are different and we can apply the general method.

/*3 (y,, zi) - ^^ (x, zl) satisfies both lines and we can proceed

with the following iteration.

- 11 -

The solutions

are

0,4, C0, 014] , 0,6 � They are belong to f0,4, 0,4,

0,6 � | 4) The values if Uop (x, z1) are not distinct

We proceed as in 2 and 3 beginning by 2 .

Conclusion : Thus the above algorithm enables us to determine the solution of fuzzy equations ; it can of course be applied to fonctional relations ; unfortunately it does not enableVto obtain necessary and sufficient conditions of solutions.

Examples :

The solution is ( 0,4, 0,4, 0,6)